First of all I do not see why we after we have expanded it in the first row why we set this less than, second row. I also do not see why we have more than one fraction with $N $ in the denominator on the second line.

I understand the objective to show that $0$ times a constant $K$ is equal to zero. But could someone take me through this step by step. I usually get induction but I'm just stuck with this one.

$\begingroup$Please do not use titles consisting only of math expressions; these are discouraged for technical reasons -- see meta.$\endgroup$
– AlexRJan 11 '15 at 22:21

$\begingroup$Note that $\lim_{n\to\infty}\sum_{k=0}^n\frac{x^k}{k!}=e^x$ (and the series converges absolutely for all $x\in\mathbb C$) so necessarily $\lim_{k\to\infty}\frac{x^k}{k!}=0$. (I know this doesn't answer your question, but just something to keep in mind.)$\endgroup$
– Math1000Jan 11 '15 at 22:28

$\begingroup$I would understand in row 2 if was as you described for the answer to my second question but it is multiple fractions with only N in the denominators. Why not ex. N-1, N, N+1$\endgroup$
– ALEXANDERJan 11 '15 at 22:03

$\begingroup$To each $N+1, N+2,\ldots, n$ you do as I showed on the asnwer, then you'll get each term of the multiplication, since $a<c \Rightarrow ab<cb$, if $b > 0$.$\endgroup$
– Aaron MarojaJan 11 '15 at 22:10

$\begingroup$And notice that the terms $1,2, \ldots,N-2,N-1$ were already use to get $(N-1)!$$\endgroup$
– Aaron MarojaJan 11 '15 at 22:15

$\begingroup$I understood the last comment as well as your edit, but I do not get why we multiply in the second line this with $\frac{\left| x\right| }{N}$*$\frac{\left| x\right| }{N}$.....,I dont see why N does not increase with 1 in each step.$\endgroup$
– ALEXANDERJan 11 '15 at 22:27